BSTJ 52: 1. January 1973: Applications for Quantum Amplifiers in Simple Digital Optical Communication Systems. (Personick, S.D.)

Copyright © 1973 American Telephone and Telegraph Company

The Bell System Technical Journal

Vol. 52, No. 1. January, 1973

Printed in U.S.A.

Applications for Quantum Amplifiers in
Simple Digital Optical Communication

Systems

By S. D. PERSONICK

(Manuscript received August 14, 1972)

Previously published results on the -performance of optical direct de-
tection digital receivers using avalanche detectors are extended to the case
where incoherent noise due to quantum amplifiers in the transmission
medium is present at the detector. These calculations are applied to de-
termine the usefulness of quantum amplifiers in simple digital transmission
systems where the optical source instability results in a required amplifier
bandwidth which may be orders of magnitude greater than the modulation
bandwidth. It is concluded that practical applications exist where quantum
amplifiers can be used in analog repeaters between regenerating repeaters in
a hybrid digital system; and also as front ends of regenerating repeaters to
increase their sensitivities.

I. INTRODUCTION

Quantum amplifiers can be used in optical communication systems
even if the optical sources are only partially coherent. They can serve
as optical analog repeaters between regenerating repeaters in a
hybrid digital system to compensate for transmission loss (see Fig. 1),
and also as the front ends of regenerating repeaters which demodulate
back to baseband.

This paper investigates the applications for quantum amplifiers
in simple digital communication systems employing "on-off" intensity
modulation. It will be assumed that due to source instability, the
optical system bandwidth may be orders of magnitude greater than
the bandwidth of the modulation, and that the quantum amplifiers
have limited gain.

We shall calculate Chernov bounds on the required signal energy per
pulse at the detector of a digital repeater (to be described in detail
below) to achieve a 10 -9 error rate as a function of the received sponta-

117

118

THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1973

SOURCE

MODULATOR

FIBER f— r —

QUANTUM
AMPLIFIER

FIBER

EQUIVALENT
NOISE

EQUIVALENT
NOISE

QUANTUM
AMPLIFIER

FIBER

REGENERATOR

FIBER

Fig. 1 — Fiber communication system.

neous emission noise level from quantum amplifiers, or of the incoherent
background noise level for some typical values of dark current of the
detector, mean detector gain, and thermal noise introduced by circuitry
following the detector. For these Chernov bound calculations, we shall
assume a unilateral gain detector. Numerical results for other detectors
and parameter values can be obtained by using the moment generating
functions to be derived below with the results of two previous papers 1 ' 2
concerning Chernov bounds for direct detection intensity modulation
systems using avalanche gain.

We shall also derive some signal-to-noise ratio results which can be
used to approximate the energy required per pulse to achieve a desired
error rate. These signal-to-noise ratio results are consistent with
previous published work of other authors. 3-5

Throughout this paper it will be assumed that the modulation
consists of varying the intensity of the transmitted signal in each
baud interval to produce pulses at the regenerating repeater which are
of one of two amplitudes, and such that the pulses are approximately
constant in a baud interval of length T seconds. Generalization to
other pulse shapes should be straightforward using the results below.

II. A MODEL FOR THE QUANTUM AMPLIFIER NOISE

Throughout this paper we shall model the source as follows. Its
voltage in a single spatial mode will be given by

E.(t) = tfre{Ae iia+uU } (1)

where \<a\ < 2tB/2.

That is, the source will be nominally at optical frequency ti/2w but
due to source instability there will be an uncertainty of bandwidth B.
(The conclusions and numerical results that follow also hold if the
source is a randomly phase-modulated sinusoid having a bandwidth B
of the form E,(t) = yJ2re{Ae i(6W+Qt) \.)

If the modulated signal is to be transmitted over a channel with
quantum amplifiers and possibly with optical filters as well, then these

QUANTUM AMPLIFIERS 119

devices must have a bandwidth of at least B + l/T to accommodate
the modulated signal for all possible values of w. (The term l/T is due
to the increase in bandwidth of the source due to the modulation.)

At the regenerating repeater input, the classical field will be modeled
as follows (assuming only a single spatial mode)

E r (t) = y/2re{m(t)e i(a +" )t + n(t)\ (2)

where m(t), the modulation, assumes one of the two possible pulse
amplitudes (a pulse which is approximately constant in each baud
interval T) and n(t) is a complex Gaussian random process which
represents the incoherent spontaneous emission noise introduced in
the quantum amplifiers or represents incoherent background noise. 6
In each baud interval T, we can expand the field complex envelope in
a Fourier series; 1 taking only enough terms to include the "system"
bandwidth B' }

-(l-d/2 L vr J

where L (the number of temporal modes) is given by L = B'T ^ 1
+ BT. Defining

m k = -7= I m(t)e iat e- i v* ktlT) dt ,

vi J baud interval

n k = -= f n(t)e-^' k 'i T >dt , (4a)

\i J band interval

we have for each value of k, a k = m k + n k . Because we have a digital
system, the signal components, m k , take on one of two values for each
k. The noise components n k are complex Gaussian random variables.

(n*n*) = N d kh (n*n y ) = (4b)

where iV is the classical incoherent noise spectral height 8 : (n(t)n*(T))
= N 8(t — t), and (x) stands for the expected value of x.

f A more rigorous and general approach taken in the Appendix is to expand the re-
ceived field in a Karhunen-Loeve expansion 5 using the autocorrelation function of the
noise n(t) at the detector input as the kernel. The approach taken here is justified on
grounds of simplicity and intuitiveness.

1 The system bandwidth B' is the minimum of the quantum amplifier bandwidth,
the detector optical bandwidth, and the bandwidths of any filters in the optical path
preceding the detector. Of course B' > B + (l/T), if we are to accommodate all
possible signals with the unstable source described above.

§ That is, the number of watts of incoherent power falling on the detector in the
bandwidth B' is X B'.

120 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1973

At the regenerating repeater it will be assumed that the field falls
upon a detector with internal gain (e.g., an avalanche detector) and
causes the detector to emit "primary" hole-electron pairs at rate
(average pairs/second)

X(0 =^-[e r (f)| 2 (5)

nil

where h = Planck's constant/271-, fi = optical frequency in radians/s,
t? = detector quantum efficiency, and e r (t) was defined in eq. (3) above.
Due to internal gain, each primary "count" (hole-electron pair)
produces a random number of additional secondary counts. Because
the modulation pulse is approximately constant throughout a baud
interval, we will be interested in the total number of counts produced
by the detector due to signal and incoherent noise in each baud interval.
The moment generating function 7 of the random total number of
counts, N, produced in each baud interval, T, is defined as

00

M N (s) = L e«»p(n) (6)

(1 =

where

p(n) = probability that N = n.

From previous work 1 we have

M N (s) - McGMa)) (7)

where

+ G (s) is ln[M <?(*)].

M (s) is the moment generating function of the random internal gain
G and M c (s) is the moment generating function of the total number of
primary counts, C.

We can evaluate M c (s) as follows. Define the quantity A as

A = — / \er(t)\*dt = — E M 2 (8)

AJ2 ./baud interval «S2 -(L-D/2

where A is the average number of received primary counts in a baud
interval given e T (t). A is a random variable, since the \a k ) are random
variables having the following joint complex Gaussian probability

density

L 1
p{a h a 2 ---a L } = II — - e" l<u " " ,kl ' /N °-
k=\ tJ\ o

The probability distribution of the total number of primary counts

QUANTUM AMPLIFIERS

121

DATA
SOURCE

"ONE"

OPTICAL
SOURCE

OPTICAL
SOURCE

"ZERO-

•• TO "ONE" CHANNEL

■*• TO "ZERO" CHANNEL

THERMAL
NOISE

pORp-EXT

FROM
CHANNELS

DETECTOR

POWER

p-EXTORp

DETECTOR

Idt

•(-1)

"ONE"

X >
— <°

"ZERO"

THERMAL
NOISE

Fig. 2 — Twin-channel system.

C in a baud interval given A is Poisson, i.e.,

A c e _A

p(c\A) = = probability that C = c given A.

c!

It follows that M c (s) is given by
M c (s) = / Lp(c|A)e" \p(A)dA

e A(e.-l) p ( A ) rfA

vNo

■L

- \ \ . - — - (e'

Xe I (•»/*«) 2 |7n*|«(«' - D/[l - (v/hil)No(e' - l]| f (9)

III. SIGNAL-TO-NOISE RATIO RESULTS

From (7) and (9) we obtain the mean number of counts, (N),
emitted by the avalanche detector in a baud interval as follows

(N) = -M N (s) = 7z Mc(Ms)) — Ms)

ds d[>c(s)] ds

= G[m 2 + LNMhQ

(10)

122 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1973

where

r (Z.-D/2

m 2 = / \m(t)\ 2 dt = £ \m h \ 2 ,

J baud interval — (L— 1)/2

No = ( | n k , 2 ) = classical spectral height of the incoherent noise
at the detector input,
L ^ \_B + l/T~\T = BT + 1,

and G is the mean avalanche gain.

The variance of the total number of counts is

V-

(N 2 ) - (N) 2 = — M N (s
ds 2

( d I y

-( — M N {8)\
o \os |,_o/

= (m* + LiV ) — G^ + (LJV. 2 + 2N m 2 ) (G) 2 (^-)

hQ \»Q/ (11)

shot noises beat noisest

where G 2 is the mean square avalanche gain.

Consider a typical twin-channel digital system, shown in Fig. 2.
There is light incident on each detector containing "on-off" modulated
signal pulses of duration T and incoherent noise. A channel is in the
"on" state when its signal pulse has optical power p. In the "off" state
the signal pulse power is p-EXT, where EXT is small compared to
unity. During each baud interval, one or the other channel is "on."
The detectors are assumed to have internal random gain (e.g., av-
alanche gain or photomultiplier gain) and there are assumed to be
thermal noises added to the detector outputs due to the amplifiers
following the detectors. It is assumed that the signaling rate is slow
enough so that each signal pulse of light of duration T produces an
output current from its detector of duration T that does not overlap
with the currents from other pulses. The detector output current
pulses plus the corresponding noises are integrated in each period T
(or equivalently filtered). The output variable x is compared to the
threshold after each integration to decide which channel is "on." An
error is made if .r > when the "zero" channel is on, or vice-versa.

The baseband noise-to-signal ratio is defined as the variance of the
output voltage x divided by the square of the mean of the output
voltage x.

* The term "bent noise" has been used in literature 8 to describe those noise terms at
the output of a square law detector which are due to fluctuations in the instantaneous
power of a carrier which has a fluctuating amplitude.

QUANTUM AMPLIFIERS 123

(x 2 ) - ((x)) 2 _ 4k0T

(x) 2 ~ Re 2 \ 2 (l - EXT) 2 G 2

thermal noise

/ -

2\ d + 2LX„ + X,(l + EXT) G 2

X 2 (l - EXT) 2 ~~(G)-<

shot noises

2/ /

2LX„ + 2X,X„(1 + EXT)

+ (12)

X 2 (l - EXT) 2

beat noises

where

k0 = Boltzman's constant • absolute noise temperature referred
to the integrator input.

R = integrator equivalent thermal noise input resistance.

Xd — mean dark current counts per detector per interval T before
avalanche gain.

X, = m 2 t]/hQ, = mean signal counts per interval T in "on" chan-
nel before avalanche gain.
LX„ = mean incoherent noise counts in either channel per baud
interval T before avalanche gain.

L ^ BT + 1, and equals the number of temporal modes de-
tected.
EXT = Signal power in "off" channel/signal power in "on" channel.

In eq. (12), terms which are due to the incoherent spontaneous
emission noises of the quantum amplifiers (or background noise) are
marked with arrows.

We see that the optical incoherent noise, when detected to baseband,
causes additional shot noise and also contributes two beat noise terms.
One of these is proportional to the signal X, and one is proportional to
L. One can use these signal-to-noise ratio results to approximate the
error rate by assuming that the output variable x is roughly Gaussian
in distribution.

In the next section Ave shall generate some curves that may give a
clearer picture of the effects of L, X„, X„ etc., on performance.

124

THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1973

IV. CHERNOV BOUNDS

The moment generating function denned in (9) was used with
previously published results 1 - 2 on avalanche photo-diode gain statistics
to obtain Chernov upper bounds on the energy per pulse required at
the input of a digital twin-channel regenerating repeater of Fig. 2 to
achieve a desired error rate as a function of the other parameters.

The general Chernov bound is given as follows. 7 Let X be a random
variable with moment generating function M x (s). Let Pr x (x > 7) be
the probability that an outcome x of X exceeds 7- Then it follows that

Pr x (x > 7) ^ «[**<•>-•*! for s > (13)

where

+ x (s) = ln[M x (s)].

The bound is optimized for s such that (6y(s)/ds) = 7 provided
that value of s is greater than zero.
Similarly,

Pr x (x < 7) ^ e [*x(.>—7] for s < (14)

where the optimal value of s is given by (d\Js(s)/ds) = 7 provided that
value of s is less than zero.

To obtain Chernov bounds upon the probability of error for the
twin-channel system of Fig. 2, one needs the moment generating

- 10 3

ERROR RATE- 10" 9
EXT -0.001

NOISE COUNTS PER MODE, X r

Fig. 3 — Required energy per pulse normalized by ri/Ml vs the incoherent noise level
N at the detector, also normalized by i}/Ml.

QUANTUM AMPLIFIERS

125

ERROR RATE

= 10" 9

4

EXT

= 0.001
-5

L= 10,000,/

2

- V-

G
kflT
Re 2

= 20
= 600

>^/ 1000-^

/

*

f

10 J
B

6
4

5-^^

/_■

S /
/

/

/

7 //~

2

~ /

/

/ //

in 2

1

I 1 1

i i iii

i

i iii

10

B ^qO ^ 1 b B ^

NOISE COUNTS PER MODE, A R

Fig. 4— Same as Fig. 3.

'10'

function of the output variable x. This can be obtained using (9) and
the results of the Refs. 1 and 2, which are too detailed to duplicate here.

From simple cases where error rates can be calculated exactly, the
differences in required power between those results and the bounds
are typically a few dB or less. Experimental results also confirm the
tightness of the bounds. Therefore, in this paper we shall take the
liberty of comparing the effects of various parameters upon the
required energy per pulse to achieve a desired error rate by comparing
the bounds.

It was decided that the calculations should be presented graphically
in two ways.

First, in Figs. 3 to 5, the required energy per pulse normalized by
T)/hQ, (i.e., the mean number of detected signal photons per pulse) is
plotted vs the incoherent noise level N at the detector also normalized
by ri/hQ.. This is done for various values shown of L, mean avalanche
gain, thermal noise, dark current, error rate, and extinction ratio, for a
low-noise unilateral gain avalanche detector (i.e., a detector in which
only one type of carrier causes ionizing collisions, and where carrier
injection is from one end of the high field region). The avalanche gains
used in these calculations do not minimize the required energy per
pulse for the given values of the other parameters, but were used for
illustration.

126 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1973

G = 100

10

I I I

10'

NOISE COUNTS PER MODE, X n

Fig. 5 — Same as Fig. 3.

4 6 8 10 2

It was recognized that in a hybrid system, if the loss between the
regenerating repeater and the analog repeater closest to it is in-
creased, then the signal energy per pulse at the regenerating repeater

S m3 _

5 10'

8 8

<3

m 4

10'

io-

-

ERROR RATE = 10 -9
EXT = 0.001

G = 4

/ 2k0T
V Re * =6 °

L= 10,000

/lOOO 1

-

/ / 10 °

_

no

- y

/'

i

I I I I

I 1 1 1 1

1 1 1 L_

NOISE COUNTS

PER MODE

SIGNAL

COUNTS

, Z

Fig. 6 — Required energy per pulse normalized by rj/iiii vs the ratio Z of spontane-
ous emission noise spectral height to signal energy.

QUANTUM AMPLIFIERS

127

10'

10"

ERROR RATE = 10~ 9

G = 20

-

EXT = 0.001

A

/2k»T

/ 5-= 600

Re 2

-

A. = 10,000

/1000 1

—

/ / 10 °

MO

: y

/I

"^

1 III

1

1 1 1

1

1 1 1 1

NOISE COUNTS

PER MODE

SIGNAL

COUNTS

10

Fig. 7 — Same as Fig. 6.

input will decrease while the ratio of signal energy per pulse to spon-
taneous emission noise spectral height at the regenerating repeater
input will remain fixed. Thus in Figs. 6 to 8, the required energy per
pulse normalized by rj/hQ is plotted vs the ratio Z of spontaneous

ERROR RATE- 10" 9
EXT = 0.001
A ri = 5

2K0T

I I I

10-

NOISE COUNTS

PER MODE

SIGNAL

COUNTS

Fig. 8 — Same as Fig. 6.

128 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1973

emission noise spectral height to signal energy, for the various values
shown of other parameters.

V. APPLICATIONS AND EXAMPLES

5.1 Analog Repeaters

Suppose one used quantum amplifiers in analog repeaters placed
between regenerating repeaters so as to increase the distance between
regenerating repeaters. See Fig. 1. Each quantum amplifier introduces
a spontaneous emission noise which has spectral height referred to its
input given by 6

(ri i\
— ) (15)

where G q is the quantum amplifier power gain and F is a noise figure
which can be near unity for good quantum amplifiers and is typically
less than 10. f If the input to analog repeater k is a k nepers (in power)
higher than the signal level at the input of the regenerating repeater,
then the total spontaneous emission noise spectral height N at the
input of the regenerating repeater is

No = £ F k hSl—^ e—* (16)

1 G q k

where R = number of analog repeaters.

The ratio of N in (16) to the signal energy per pulse, p-T at the
regenerating repeater input, see Fig. 2, is the parameter Z defined in
Section IV above. Since the incoherent noise and the signal both
experience equal loss per unit length from the fiber, the ratio Z is
constant between the regenerating repeater and the analog repeater
closest to it.

Example: Suppose we make the following assumptions. A twin-channel
system is used with a unilateral gain detector having mean gain 100
and with all the other parameter values necessary above so that the
Chernov bound curves of Fig. 8 are applicable. The source is a
Nd:YAlG laser having bandwidth 1 A at wavelength 1 /im, i.e.,
3 • 10 10 Hz. The modulation rate is 300 Mb/s so that T ~ 3.33 X 10" 9 s.

t F is related to the population inversion in the amplifying medium which is as-
sumed constant in this analysis.

QUANTUM AMPLIFIERS 129

We then have L = 100. There are 10 analog repeaters and they are
spaced so that the signal level is the same at the input to each one.

From Fig. 8 (assuming that the upper bounds are tight enough so
that we can comment upon the effects of various parameters on the
required energy per pulse by observing their effects upon the bounds*)
we see that when Z is less than 10 -3 , the required signal energy at the
regenerator input is 600 counts, i.e., p-T = hft/r)- GOO. This value of
signal energy is the same as that which would be required if no sponta-
neous emission noise were present (Z = 0).

Thus for spontaneous emission noise to be negligible in this example,
we must have the ratio of the signal energy per pulse at the regenerator
input to N larger than 10 3 .

This means [from (16)3 that at the analog repeater inputs the signal
level must exceed 10 3 -tiSlRFKGg- 1) /GJ where

R = number of analog repeaters = 10 in this example
G q = gain of analog repeater (assumed the same for all repeaters)
F = noise figure of an analog repeater (assumed the same for all
repeaters).

Looking again at Fig. 8, we see that for L = 100, Z can be as large
as 5 X 10~ 3 before the required signal level at the regenerating repeater
becomes large and enters the sensitive region. This means that the
signal level at the inputs to the analog repeaters might be as low as
200 • hQBF[_(6 t — 1)/Gq} in which case the signal required at the
regenerating repeater is somewhat larger, but still not extremely
sensitive to small changes in Z. Suppose F = 10, 77 ^ 1, G q = 100, and
the maximum power output of any repeater is 1 mW. Suppose the loss
of the medium is 10 dB/km. When spontaneous emission noise is
negligible, we need 600 hSl = 1.2 X 10 -16 joules per pulse at the input
to the regenerating repeater and we have 3.33 X 10 -12 joules per pulse
at the output. Without analog repeaters we can have about 44.5 dB of
loss or 4.45 km between regenerating repeaters. Suppose on the other
hand we use 10 analog repeaters starting where the signal level is 200
hSlRF£(G q - \)/G q ~\ = 4 X 10~ 15 joules per pulse (i.e., Z = 5 X 10~ 3 );
or about 28.8 dB (2.88 km) from the regenerating repeater output.
The string of 10 analog repeaters spaced at 20-dB intervals spans 200
dB or 20 km of distance ; and we can have an additional 13 dB or 1.3 km
of distance to the next regenerating repeater input resulting in the
required 2 X 10~ 16 joules per pulse at that regenerating repeater input.

t See comment Section IV.

130 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1973

The total distance between regenerating repeaters is now about 24.2

km. +

It seems prudent that for a given value of L, one should avoid values
of Z which are so large that small changes in Z result in large changes
in the required signal energy at the regenerating repeater. Such small
changes in Z might come about if the source power or quantum ampli-
fier gains fluctuated slightly.

5.2 Regenerator Repeater Front End

Suppose that in the example above we had just one quantum
amplifier (or equivalently, the spontaneous emission noise from any
additional quantum amplifiers was negligible).

In the absence of spontaneous emission noise, the energy per pulse
required at the regenerative repeater input is approximately GOO ftfl/ij.
Now suppose we place the quantum amplifier immediately before the
regenerating repeater. If the gain is sufficiently large, then we can
operate with Z as large as 7 ■ lO" 3 . This means the energy per pulse at
the input to the quantum amplifier need only be about h$lF/(7 X 10 -3 )
« UOhSlF (for large G q ). Thus, we see that if 140/' 1 < 600/tj, then the
quantum amplifier increases the sensitivity of the regenerative repeater
over that associated with an avalanche detector alone (in this example
with L = 100).

For other values of L in this example, the condition for a quantum
amplifier front end to increase the regenerative repeater sensitivity is

F 600
<

Zmax 77

where Z max is the maximum value of Z for reasonable required energy
per pulse at the input to the regenerating repeater (following the
quantum amplifier).

For other systems with different types of avalanche detectors and
different parameters (avalanche gain, dark current, etc.) the number
600 in the above equation should be replaced by the required mean
number of detected counts in the absence of a quantum amplifier.

t A slightly larger total distance between regenerating repeaters can be obtained by
starting the chain of analog repeaters 20 dB (rather than 28.8 dB) from the regener-
ating repeater output. In that case Z ^ 5-10 - * and the next regenerating repeater can
be about 45 dB from the last analog repeater for a total span of 24.5 km between re-
generating repeaters. Placing the analog repeaters as described in the above example
allows some margin for overload.

QUANTUM AMPLIFIERS 131

5.3 Background Noise

As a final comment, it is clear from eq. (12) that if the incoherent
noise spectral height, N„, at the regenerating repeater input is small
enough so that (n/fiQ)N « G 2 /(G)- then only the additional shot noise
term is important amongst the three noise terms associated with the
incoherent noise.

This inequality always holds for the case where the incoherent
noise is background (thermal) radiation in equilibrium at temperatures
below 10 4 °K, since for thermal background radiation we have

N (Thermal) =

gfcn/ke _ i '

kd = Boltzman's constant -absolute temperature.

At room temperature and at a wavelength of 1 fxm, hQ,/kd ~ 50.

Therefore, in analyses where incoherent background radiation is
included, one usually only includes the additional shot noise term
LN (r)/hti) = L\„ in the signal-to-noise ratio formulae.

VI. CONCLUSIONS

Wc have shown that quantum amplifiers can have applications in
both analog repeaters to extend the distance between regenerating
repeaters and as front ends of regenerating repeaters. Their usefulness
is a function of the ratio of the optical bandwidth of the system to the
modulation bandwidth; but is not limited to small values of this ratio.
To choose system parameters, for example, the required signal levels
at the analog and regenerating repeater inputs, various component
parameters such as the mean avalanche gain, avalanche detector type,
source bandwidth, baseband thermal noise, etc., must be given.
Computations in addition to those presented, upper bounding the
error rates, can be carried out with previously published Chernov
bound results; 1 ' 2 or approximate error-rate calculations can be made
using the signal-to-noise ratio results of Section III above.

APPENDIX

Use of the Karhunen-Loeve Expression

Starting with eq. (2) of the text, we could expand the received
complex envelope e r (t) = m(t)e' ut + n(t) in a baud interval in terms
of the Karhunen-Loeve eigenf unctions of the band limited incoherent

132 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1973

noise n(t), i.e., define

R n (t,u), [Mu)}, and (7*}

as follows

R n (t,u) = (n(t)n*(u))

/ \f/ k (u)R„(t,u)du = y^k(t) for U baud interval,

J baud interval

A: = 1,2,3

Then

where

*r(t) =

i

a k \p k (t)

for

te(0,T)

a k =

m k

+ n k

m k =

1 ?r

J baud interval

(t)e {

"Mt)dt

n k

= J

(

baud interval

n(t)ih(t)dt

and

(n k nj) = y k 8 k j, (n k rij) =

/ yp k (t)^{t)dt = 8 kj .

Jbi

'baud interval

Then we would find that M c (s) of eq. (9) could be more rigorously
given by

Thus in eq. (9) N has been rigorously replaced by y k for each k
and the finite number of terms L has been replaced by an infinite
number of terms.

If we make the reasonable assumption that the incoherent noise
is Hat with spectral height N in a band of width B' + l/T then

7* « No for 1 ^ ft ^ L

« otherwise (17)

QUANTUM AMPLIFIERS 133

where

L = B'T + 1.

Thus the form for M c (s) derived in the main text is identical to the
more rigorous result under this approximation.

REFERENCES

1. Personick, S. D., "New Results on Avalanche Multiplication Statistics with Appli-

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